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Various glossy difficulties in physics, chemistry, and quantum electronics require a attention of inhabitants dynamics in complicated multilevel quantum platforms. the aim of this publication is to supply a scientific therapy of those questions and to provide a couple of precisely solvable difficulties.

29). Therefore they are identified as a set of creation-annihilation operators creating states with well-defined momentum p out of the vacuum |0 . In the canonical quantization formalism the concept of particle appears as a result of the quantization of a classical field. From the expressions of φˆ and πˆ in terms of the creation-annihilation operators we can evaluate the Hamiltonian operator. After a simple calculation one arrives at 1 Hˆ = 2 = d3 p αˆ † (p)α(p) ˆ + (2π )3 E p δ(0) (2π )3 1 d3 p 1 E p αˆ † (p)α(p) ˆ + d 3 pE p δ(0).

The lesson we have learned in quantum mechanics can now be used to study the semiclassical tunneling between different vacua in a field theory by means of an analytic continuation to imaginary times. Letting t → −it, the Minkowski spacetime transforms in Euclidean space ds 2 = ημv d x μ d x v =⇒ ds 2 = −δμv d x μ d x v . 102) In the example of the scalar field theory discussed above, this analytical continuation to imaginary time leads to the Euclidean action S E [φ] = d4x 1 μv δ ∂μ φ∂v φ + V (φ) .

86) is dominated by paths around the classical solution qcl (t) that solves the equations of motion with the appropriate boundary conditions. Thus, the propagator K (q, q ; τ ) can be computed in the semiclassical limit by considering only the contribution to the path integral coming from paths that are “close” to the classical one. 5 The Semiclassical Limit 29 and integrate over the perturbation δq(t) keeping in the action only terms that are at most linear in τ S[q] = Scl [q, q ; τ ] + dt m(δ q) ˙ 2 − V (qcl )(δq)2 + O( 3 ).